Let the length of shortest side = ‘x’ cm
According to the question,
The longest side of a triangle is twice the shortest side
⇒ Length of largest side = 2x
Also, the third side is 2 cm longer than the shortest side
⇒ Length of third side = \[\left( x\text{ }+\text{ }2 \right)\text{ }cm\]
Perimeter of triangle = sum of three sides
\[\begin{array}{*{35}{l}}
=\text{ }x\text{ }+\text{ }2x\text{ }+\text{ }x\text{ }+\text{ }2 \\
=\text{ }4x\text{ }+\text{ }2\text{ }cm \\
\end{array}\]
Now, we know that,
Perimeter is more than 166 cm
\[\begin{array}{*{35}{l}}
\Rightarrow ~4x\text{ }+\text{ }2\text{ }\ge \text{ }166 \\
\Rightarrow ~4x\text{ }\ge \text{ }164 \\
\Rightarrow ~x\text{ }\ge \text{ }41 \\
\end{array}\]
Hence, minimum length of the shortest side should be = 41 cm.